A generalization of Little's Theorem on Pfaffian orientations

Little (1975) [12] showed that, in a certain sense, the only minimal non-Pfaffian bipartite matching covered graph is the brace K"3","3. Using a stronger notion of minimality than the one used by Little, we show that every minimal non-Pfaffian brick G contains two disjoint odd cycles C"1 and C"2 such that the subgraph G-V(C"1@?C"2) has a perfect matching. This implies that the only minimal non-Pfaffian solid matching covered graph is the brace K"3","3. (A matching covered graph G is solid if, for any two disjoint odd cycles C"1 and C"2 of G, the subgraph G-V(C"1@?C"2) has no perfect matching. Solid matching covered graphs constitute a natural generalization of the class of bipartite graphs, see Carvalho et al., 2004 [5].)

[1]  Robin Thomas,et al.  Generating bricks , 2007, J. Comb. Theory, Ser. B.

[2]  Claudio L. Lucchesi,et al.  On a Conjecture of Lovász Concerning Bricks: II. Bricks of Finite Characteristic , 2002, J. Comb. Theory, Ser. B.

[3]  Claudio L. Lucchesi,et al.  A Polynomial Time Algorithm for Recognizing Near-Bipartite Pfaffian Graphs , 2008, Electron. Notes Discret. Math..

[4]  Ilse Fischer,et al.  A Characterisation of Pfaffian Near Bipartite Graphs , 2001, J. Comb. Theory, Ser. B.

[5]  William McCuaig,et al.  Pólya's Permanent Problem , 2004, Electron. J. Comb..

[6]  Claudio L. Lucchesi,et al.  On the number of dissimilar pfaffian orientations of graphs , 2005, RAIRO Theor. Informatics Appl..

[7]  László Lovász,et al.  Matching structure and the matching lattice , 1987, J. Comb. Theory, Ser. B.

[8]  W. T. Tutte.  The Factorization of Linear Graphs , 1947 .

[9]  Robin Thomas,et al.  PERMANENTS, PFAFFIAN ORIENTATIONS, AND EVEN DIRECTED CIRCUITS , 1997, STOC 1997.

[10]  J. A. Bondy,et al.  Graph Theory , 2008, Graduate Texts in Mathematics.

[11]  László Lovász,et al.  Brick decompositions and the matching rank of graphs , 1982, Comb..

[12]  F. Rendl,et al.  Operations preserving the pfaffian property of a graph , 1991, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[13]  Robin Thomas,et al.  Minimally non-Pfaffian graphs , 2008, J. Comb. Theory, Ser. B.

[14]  C. Little.  A characterization of convertible (0, 1)-matrices , 1975 .

[15]  P. W. Kasteleyn.  Dimer Statistics and Phase Transitions , 1963 .

[16]  Mihalis Yannakakis,et al.  Pfaffian orientations, 0-1 permanents, and even cycles in directed graphs , 1989, Discret. Appl. Math..

[17]  Claudio L. Lucchesi,et al.  Graphs with independent perfect matchings , 2005, J. Graph Theory.

[18]  László Lovász,et al.  Ear-decompositions of matching-covered graphs , 1983, Comb..

[19]  Claudio L. Lucchesi,et al.  Optimal Ear Decompositions of Matching Covered Graphs and Bases for the Matching Lattice , 2002, J. Comb. Theory, Ser. B.

[20]  Claudio L. Lucchesi,et al.  On a Conjecture of Lovász Concerning Bricks: I. The Characteristic of a Matching Covered Graph , 2002, J. Comb. Theory, Ser. B.

[21]  Claudio L. Lucchesi,et al.  How to build a brick , 2006, Discret. Math..

[22]  Claudio L. Lucchesi,et al.  The perfect matching polytope and solid bricks , 2004, J. Comb. Theory, Ser. B.